Subgroups — #33004523

Order 18688.

350 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#17988717	C₂	1	1	yes	maximal, characteristic
2	#22611687	C₂	1	1	yes	maximal, characteristic
3	#22608014	C₂	1	4	yes	maximal
4	C₄×M₃₂×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₁₄₆	C₄	1	1	yes	characteristic
6	C₂×#14556531	C₂²	1	2	yes	—
7	#17988717	—	2	1	no	—
8	#17988717	—	2	1	no	—
9	#17988717	—	2	1	no	—
10	#17988717	—	2	1	no	—
11	C₂×C₄⋊#10596831	C₄	1	1	yes	characteristic
12	C₂×#14556520	C₄	1	2	yes	—
13	C₂×#14556501	C₂²	1	2	yes	—
14	C₄⋊#11924003	—	2	1	no	—
15	C₄⋊#11924003	—	2	1	no	—
16	C₂×M₃₂×C₇₃	C₂²	1	2	yes	—
17	C₂×C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
18	#346×C₇₃	—	2	1	no	—
19	C₄×C₁₆×C₇₃	—	2	1	no	—
20	C₂²×Dic₁₄₆	C₈	1	1	yes	characteristic
21	C₂²×#10596831	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₄₆	—	2	1	no	—
23	C₄×Dic₁₄₆	—	2	1	no	—
24	C₄×Dic₁₄₆	—	2	1	no	—
25	C₄×Dic₁₄₆	—	2	1	no	—
26	C₂×C₄⋊Dic₇₃	C₈	1	1	yes	characteristic
27	C₂×#11924007	C₈	1	2	yes	—
28	C₂×C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	#14556531	—	2	4	no	—
30	C₂×#11924003	—	2	2	no	—
31	C₄⋊#10596831	—	2	1	no	—
32	C₄⋊#10596831	—	2	1	no	—
33	#14556520	—	2	2	no	—
34	#14556520	—	2	2	no	—
35	#14556501	—	2	2	no	—
36	#14556501	—	2	2	no	—
37	C₂²×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈×C₇₃	C₂³	1	1	yes	characteristic
39	C₂×C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
40	M₃₂×C₇₃	—	2	2	no	—
41	M₃₂×C₇₃	Q₈	1	4	yes	complemented
42	C₂×C₁₆×C₇₃	—	2	2	no	—
43	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
44	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
45	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
46	C₄×C₈×C₇₃	D₄	1	1	yes	characteristic
47	C₂×Dic₁₄₆	—	2	1	no	—
48	C₂×Dic₁₄₆	—	2	1	no	—
49	C₂×Dic₁₄₆	—	4	2	no	—
50	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
51	C₂²×Dic₇₃	C₂×C₈	1	2	yes	—
52	C₂×#10596831	—	2	2	no	—
53	C₂×#10596831	—	2	4	no	—
54	C₄⋊Dic₇₃	—	2	1	no	—
55	C₄⋊Dic₇₃	—	2	1	no	—
56	#11924007	—	2	2	no	—
57	#11924007	—	2	2	no	—
58	C₄×Dic₇₃	—	2	2	no	—
59	C₄×Dic₇₃	—	2	2	no	—
60	#11924003	—	4	2	no	—
61	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
62	C₂²×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
63	C₂²×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₇₃	—	2	2	no	—
65	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₇₃	C₂×Q₈	1	2	yes	—
70	C₂×C₈×C₇₃	#41	1	2	yes	—
71	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
72	C₄²×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
74	C₄²×C₇₃	M₁₆	1	1	yes	characteristic
75	C₁₆×C₇₃	—	2	4	no	—
76	C₂³×C₇₃	C₂²×C₈	1	1	yes	characteristic
77	C₂×C₄×C₇₃	—	2	2	no	—
78	C₂×C₄×C₇₃	—	4	1	no	—
79	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
80	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
81	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
82	C₂×C₄×C₇₃	C₂²⋊C₈	1	1	yes	characteristic
83	C₂×C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₇₃	C₄×Q₈	1	2	yes	—
85	C₂×C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
86	C₂×C₄×C₇₃	#131	1	2	yes	—
87	C₂×Dic₇₃	—	2	2	no	—
88	C₂×Dic₇₃	—	2	2	no	—
89	C₂×Dic₇₃	—	4	2	no	—
90	C₂×Dic₇₃	—	4	2	no	—
91	Dic₁₄₆	—	4	2	no	—
92	Dic₁₄₆	—	4	2	no	—
93	C₈×C₇₃	—	2	1	no	—
94	C₈×C₇₃	—	2	1	no	—
95	C₈×C₇₃	C₂²⋊Q₈	1	2	yes	—
96	C₈×C₇₃	#124	1	1	yes	characteristic
97	C₈×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
98	#10596831	—	4	2	no	—
99	C₂²×C₇₃	—	2	1	no	—
100	C₂²×C₇₃	—	2	1	no	—
101	C₂²×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
102	C₂²×C₇₃	C₈×Q₈	1	2	yes	—
103	C₄×C₇₃	—	2	1	no	—
104	C₄×C₇₃	—	2	1	no	—
105	C₄×C₇₃	—	2	1	no	—
106	C₄×C₇₃	#349	1	2	yes	—
107	C₄×C₇₃	#391	1	1	yes	characteristic
108	C₄×C₇₃	#408	1	1	yes	characteristic
109	C₄×C₇₃	#446	1	2	yes	—
110	Dic₇₃	—	4	2	no	—
111	Dic₇₃	—	4	2	no	—
112	#13268	—	73	1	no	maximal, Sylow
113	C₂×C₇₃	—	2	2	no	—
114	C₂×C₇₃	#1984	1	1	yes	characteristic
115	C₂×C₇₃	C₂×#349	1	1	yes	characteristic
116	C₂×C₇₃	#2115	1	1	yes	characteristic
117	C₂×#446	—	73	1	no	—
118	#2150	—	73	1	no	—
119	C₄×M₃₂	D₇₃	1	1	yes	characteristic
120	#1377	—	73	4	no	—
121	C₇₃	#13268	1	1	yes	characteristic, complemented, Sylow
122	C₂×C₄×Q₈	—	73	1	no	—
123	C₂×M₃₂	—	73	2	no	—
124	C₂×M₃₂	C₂×D₇₃	1	2	yes	—
125	#446	—	146	1	no	—
126	#446	—	146	1	no	—
127	#446	—	146	1	no	—
128	#446	—	146	1	no	—
129	C₂×C₄⋊C₈	—	73	2	no	—
130	C₂×C₄⋊C₈	—	73	1	no	—
131	C₂×#97	—	73	2	no	—
132	C₂×C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
133	C₄⋊C₁₆	—	146	1	no	—
134	C₄⋊C₁₆	—	146	1	no	—
135	#346	—	2	1	no	—
136	C₄×C₁₆	—	2	1	no	—
137	C₂²×Q₈	—	73	1	no	—
138	C₂²×C₈	—	73	2	no	—
139	C₂²×C₈	C₄×D₇₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₇₃	1	1	yes	characteristic
141	C₄×Q₈	—	146	1	no	—
142	C₄×Q₈	—	146	1	no	—
143	C₄×Q₈	—	146	1	no	—
144	C₄×Q₈	—	146	1	no	—
145	C₂×C₄⋊C₄	—	73	2	no	—
146	C₂×C₄⋊C₄	—	73	1	no	—
147	C₂×C₄²	—	73	2	no	—
148	C₂×C₄²	C₄×D₇₃	1	1	yes	characteristic
149	M₃₂	—	2	2	no	—
150	M₃₂	—	146	4	no	—
151	M₃₂	Dic₁₄₆	1	4	yes	complemented
152	C₂×C₁₆	—	2	2	no	—
153	C₂×C₁₆	—	146	2	no	—
154	C₄⋊C₈	—	146	2	no	—
155	C₄⋊C₈	—	146	2	no	—
156	C₄⋊C₈	—	146	1	no	—
157	C₄⋊C₈	—	146	1	no	—
158	#97	—	146	2	no	—
159	#97	—	146	2	no	—
160	C₄×C₈	D₂₉₂	1	1	yes	characteristic
161	C₄×C₈	D₂₉₂	1	1	yes	characteristic
162	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
164	C₂×Q₈	—	292	2	no	—
165	C₂×Q₈	—	146	1	no	—
166	C₂×Q₈	—	146	1	no	—
167	C₂²×C₄	—	73	2	no	—
168	C₂²×C₄	—	73	2	no	—
169	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
170	C₂²×C₄	C₈×D₇₃	1	1	yes	characteristic
171	C₂²×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	146	2	no	—
174	C₂×C₈	—	146	4	no	—
175	C₂×C₈	#11924005	1	1	yes	characteristic
176	C₂×C₈	#11924005	1	1	yes	characteristic
177	C₂×C₈	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
178	C₂×C₈	C₂×D₂₉₂	1	1	yes	characteristic
179	C₂×C₈	C₂×Dic₁₄₆	1	2	yes	—
180	C₂×C₈	#11924028	1	2	yes	—
181	C₄⋊C₄	—	146	2	no	—
182	C₄⋊C₄	—	146	2	no	—
183	C₄⋊C₄	—	146	1	no	—
184	C₄⋊C₄	—	146	1	no	—
185	C₄²	—	146	2	no	—
186	C₄²	—	146	2	no	—
187	C₄²	#11924005	1	1	yes	characteristic
188	C₄²	#11924005	1	1	yes	characteristic
189	C₄²	#11924011	1	1	yes	characteristic
190	C₄²	#11924011	1	1	yes	characteristic
191	C₁₆	—	2	4	no	—
192	C₁₆	—	292	2	no	—
193	C₂³	C₂×C₈×D₇₃	1	1	yes	characteristic
194	Q₈	—	292	2	no	—
195	Q₈	—	292	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	292	2	no	—
199	C₂×C₄	—	292	2	no	—
200	C₂×C₄	—	146	2	no	—
201	C₂×C₄	—	146	2	no	—
202	C₂×C₄	#14556503	1	1	yes	characteristic
203	C₂×C₄	#14556503	1	1	yes	characteristic
204	C₂×C₄	#14556503	1	1	yes	characteristic
205	C₂×C₄	#14556503	1	1	yes	characteristic
206	C₂×C₄	C₂×#11924005	1	1	yes	characteristic
207	C₂×C₄	C₄×Dic₁₄₆	1	2	yes	—
208	C₂×C₄	C₂×#11924011	1	1	yes	characteristic
209	C₂×C₄	#14556597	1	2	yes	—
210	C₈	—	2	1	no	—
211	C₈	—	2	1	no	—
212	C₈	—	292	2	no	—
213	C₈	#14556570	1	2	yes	—
214	C₈	#14556579	1	1	yes	characteristic
215	C₈	#14556591	1	1	yes	characteristic
216	C₂²	—	2	1	no	—
217	C₂²	—	2	1	no	—
218	C₂²	C₂×#14556503	1	1	yes	characteristic
219	C₂²	C₈×Dic₁₄₆	1	2	yes	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	2	1	no	—
223	C₄	—	292	2	no	—
224	C₄	—	292	2	no	—
225	C₄	#17988368	1	2	yes	—
226	C₄	#17988493	1	1	yes	characteristic
227	C₄	#17988553	1	1	yes	characteristic
228	C₄	#17988717	1	2	yes	—
229	C₂	—	2	2	no	—
230	C₂	#22610867	1	1	yes	characteristic
231	C₂	C₂×#17988368	1	1	yes	characteristic
232	C₂	#22611551	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11924005
commutator, prime radical	C₂³×C₇₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₇₃
Fitting	C₄×M₃₂×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₈
cosocle	C₂²×C₈×C₇₃	C₂³
hypercenter	C₄×M₃₂	D₇₃
hypocenter	C₇₃	#13268
LCS	G ▷ C₂³×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ C₄×M₃₂

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
41. M₃₂×C₇₃	Q₈	semidirect
121. C₇₃	#13268	semidirect
151. M₃₂	Dic₁₄₆	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#17988717	2
#22611687	2
#22608014	2
C₄×M₃₂×C₇₃	2
#13268	73